The exclusive γE̸T signal has a small standard model cross section and is thus a channel sensitive to new physics. This signature is predicted by models with a superlight gravitino or with large extra spatial dimensions. We search for such signals at the Collider Detector at Fermilab, using 87   pb−1 of data at s√=1.8   TeV, and extract 95% C.L. limits on these processes. A limit of 221 GeV is set on the scale |F|1/2 in supersymmetric models. For 4, 6, and 8 extra dimensions, model-dependent limits on the fundamental mass scale MD of 0.55, 0.58, and 0.60 TeV, respectively, are found. We also specify a “pseudo-model-independent” method of comparing the results to theoretical predictions.